This condition is a regularization version of Condition REF , and provides restrictions on the relative divergent speeds of \(S_n\) , \(\xi _n^{R}\) , \(q_{\max }\)  and \(d_n\) . Compared with Condition REF , Condition REF  concerns \(q_{\max }\)  rather than \(p_{\max }\) , and it is likely to hold if \(q_{\max }\)  grows at a slow speed. Note that \(q_s\)  is related with the “variable selection consistency", an important property of Lasso-type estimators [1]}, [2]}.
